1
RANDOMIZED COMPLETE BLOCK DESIGNS (Chapter 10)
Introduction to Blocking
Nuisance factor: A factor that probably has an effect on the response, but is not a factor that we are interested in.
Types of nuisance factors and how to deal with them in designing an experiment:
Characteristics Examples How to treat Unknown, uncontrollable Experimenter bias, effect of order Randomization of treatments Known, uncontrollable, IQ, weight, previous learning Analysis of measurable Covariance Known, moderately Temperature, location, time, Blocking controllable (by choosing batch, particular machine or rather than adjusting) operator, age, gender, order, IQ, weight
Randomization can in principal be used to take into account factors that can be treated by blocking, but blocking usually results in smaller error variance, hence better estimates of effect. Thus blocking is sometimes referred to as a method of variance reduction design.
The intuitive idea: Run in parallel a bunch of experiments on groups of units that are fairly similar. These are called blocks.
The simplest block design: The randomized complete block design (RCBD) v treatments (They could be treatment combinations.) b blocks of v units each; blocks chosen so that units within a block are alike (or at least similar) and units in different blocks are substantially different. (Thus the total number of experimental units is n = bv.) The v experimental units within each block are randomly assigned to the v treatments. (So each treatment is assigned one unit per block.) Note that experimental units are assigned randomly only within each block, not overall. Thus this is sometimes called a restricted randomization.
Example: Five varieties of wheat are to be compared to see which gives the highest yield. Eight plots of farmland are available for the experiment. The experimenter divides each plot into five subplots. For each of the 8 plots, the varieties of wheat were randomly assigned to the subplots of that plot. Treatment factor = Response = Blocking factor = Blocks = Experimental units = v = b = n = #exp units = 2
The RCBD Model: Yhi = µ + θh + τi+ εhi 2 εhi ~ N(o,σ ) εhi’s independent where Yhi is the random variable representing the response for treatment i observed in block h, µ is a constant (which may be thought of as the overall mean – see below) th θh is the (additive) effect of the h block (h = 1, 2, … , b) th τi is the (additive) effect of the i treatment (i = 1, 2, … , v) th th εhi is the random error for the i treatment in the h block. (Why is there no subscript t for observation number?)
Note: • This model is also called the block-treatment model. Formally, it looks just like a two-way main effects model – but you need to remember that there is just one factor plus one block, and the randomization is just within each block. So we don’t have the conditions for a two-way analysis of variance. • Like the main-effects model, this is an additive model that does not provide for any interaction between block and treatment level – it assumes that treatments have the same effect in every block, and the only effect of the block is to shift the mean response up or down. If interaction between block and factor is suspected, then either a transformation is needed to remove interaction before using this model, or a design with more than one observation per block-treatment combination must be used. (Trying to add an interaction term in the RCBD would create the same problem as is encountered in two-way ANOVA with one observation per cell: the degrees of freedom for the error are zero, so the method of analysis breaks down. So if interaction is suspected, block size must be increased to allow an estimation of interaction.) b v
• This is an over-specified model; the additional constraints #"h = 0 and #" i = 0, h=1 i=1 are typically added, so that the treatment and block effects are thought of as deviations from the overall mean. • There is an alternate means model Yhi = µih + εhi, where µih = µ + θh + τi. b ! ! th 1 • Note that the i treatment mean is µi = $(µ + "h + # i ). Assuming the constraint b h=1 b
h = 0, this gives µi = µ + τi h=1 ! Estimating and Analysis:
Least squares fits: Since the model is formally the same as the main-effects model, the process of finding least squares is the same as for the latter model, yielding estimates (with notation appropriately changed) 3
µ^ = y ••
θh^ = y h• y ••
τi^ = y •i y •• y ^ = ^ + ^ + ^ = hi µ θh τ = y + y y h• •i ••
Thus the error sum of squares for this model is
ss E =
As with the two-way main effects model, msE = ssE/(b-1)(v-1) is an unbiased estimator of σ2. (Note that (b-1)(v-1) = bv – b – v + 1 = n – b – v + 1, since n = bv.)
Model checking: Look especially for potential problems with the normality assumption, unequal error variance by block or treatment, and treatment-block interaction. To check for possible block-treatment interaction, form an “interaction plot” by plotting each yhi against each treatment level i and connecting points for each block h. If corresponding line segments are parallel, this suggests both no interaction and small error variability. (Note that since there is one observation per (block, treatment level) combination, there is no way to check the equal variance assumption at that fine a level.)
Hypothesis test and Analysis of Variance Table: We are interested in testing equality of treatment means. Thus we wish to test the null hypothesis H0: µ1 = µ2 = … = µv against the alternate Ha: µi ≠ µj for at least one pair i,j.
Note: We can restate the hypotheses as H0: τ1 = τ2 = … = τv = 0 and Ha: at least one τi ≠ 0.
We can construct an F-test in the usual manner: We consider the submodel corresponding to the null hypothesis, namely
Yhi = µ + θh + εhi
This has least squares fits
(yhi^)0 = ( y ••) + ( y h• " y ••) = y h• and hence error sum of squares
! ssE0 = 4
The difference ssT = ssE0 – ssE is called the sum of squares for treatment. Our test statistic for H0 is
ssT v 1 ( ) . ssE /(b 1)(v 1) As usual, the numerator is denoted msT (with v-1 degrees of freedom) and the denominator msE (with (b-1)(v-1) degrees of freedom, as mentioned above. The test statistic has an F distribution with v-1 degrees of freedom in the numerator, (b-1)(v-1) in the denominator.
Note: • The above test is the same as the F-test for the treatment factor we would get by two-way ANOVA considering treatment and block as two factors in a main effects model. Thus we can test our hypothesis by using a two-way ANOVA main-effects software routine. • We can define ssB and msB (using b-1 degrees of freedom), but we don’t get a legitimate F-test for the null hypothesis “No block effect,” since the conditions for proving that the would-be test statistic has an F-distribution are not met, because the blocks are chosen, not randomly assigned. • Nonetheless, the ratio msB/msE can be considered as an informal measure of the effect of the blocking factor – if the ratio is large, that suggests that the blocking “factor” has a large effect, and that the variance reduction obtained by blocking was probably helpful by improving the precision in the comparison of treatment means. • The algebra works out to show that ssTot = ssB + ssT + ssE, and the degrees of freedom add accordingly.
Contrasts: In the RCBD, all contrasts (with coefficient sum zero) in the treatment effects τi are estimable, and the techniques of Chapter 4 still apply, with the following observed:
• The estimate of τi is τi^ = y •i y ••, but since in a contrast ∑ciτi,, we have ∑ci
= 0, the estimate of the contrast is ∑ci y •i • The number of replicates is equal to the number b of blocks • The error degrees of freedom are (b-1)(v-1). • The msE used is the one obtained by the block design analysis. (Thus the Minitab automatic procedur es will not work for a block design.)